Birkhoff normal forms for the (secular) planetary problem are
investigated. Existence and uniqueness is discussed and it is shown that
the classical Poincaré variables and the
ʀᴘs-variables (introduced in [6]), after a
trivial lift, lead to the same Birkhoff normal form; as a corollary
the Birkhoff normal form (in Poincaré variables) is degenerate at
all orders (answering a question of M. Herman). Non-degenerate
Birkhoff normal forms for partially and totally reduced cases are
provided and an application to long-time stability of secular
action
variables (eccentricities and inclinations) is discussed.